Positive definite matrix problems

MrJava

New member
Can anyone give me some hints for this question please?
Suppose A is positive definite and symmetric. Prove that all the eigenvalues of A are positive. What can you say of these eigenvalues if A is a positive semi definite matrix?

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Klaas van Aarsen

MHB Seeker
Staff member
Can anyone give me some hints for this question please?
Suppose A is positive definite and symmetric. Prove that all the eigenvalues of A are positive. What can you say of these eigenvalues if A is a positive semi definite matrix?

Welcome to MHB, MrJava!

Can you apply the spectral theorem for (real) symmetric matrices?

MrJava

New member
Welcome to MHB, MrJava!

Can you apply the spectral theorem for (real) symmetric matrices?
My arguments for the problem is as following:
A is positive definite matrix with λ is an arbitrary eigenvalue of A
x is a non-zero eigenvector corresponding to λ
Then:
0<xTAx=xTλx=λ(xTx) and since x'x>0, results λ>0.

Klaas van Aarsen

MHB Seeker
Staff member
My arguments for the problem is as following:
A is positive definite matrix with λ is an arbitrary eigenvalue of A
x is a non-zero eigenvector corresponding to λ
Then:
0<xTAx=xTλx=λ(xTx) and since x'x>0, results λ>0.
That's pretty close!
Mostly missing supporting arguments and slightly inconsistent.

Can you provide arguments for your reasoning?
Btw, did you do anything with the spectral theorem? Is it known to you?

MrJava

New member
That's pretty close!
Mostly missing supporting arguments and slightly inconsistent.

Can you provide arguments for your reasoning?
Btw, did you do anything with the spectral theorem? Is it known to you?
Sorry, but It seems straight to me.

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MarkFL

Staff member
Just for clarification, does this mean Senior, Señor, Sister, or perhaps Seaman Recruit, or something else?

MrJava

New member
Just for clarification, does this mean Senior, Señor, Sister, or perhaps Seaman Recruit, or something else?
Oops, I meant "Sorry"

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I am searching around the Internet and I found this on Wiki:

Let M be an n × n Hermitian matrix. The following properties are equivalent to M being positive definite:
All its eigenvalues are positive. Let P−1DP be an eigendecomposition of M, where P is a unitary complex matrix whose rows comprise an orthonormal basis of eigenvectors of M, and D is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues. The matrix M may be regarded as a diagonal matrix D that has been re-expressed in coordinates of the basis P. In particular, the one-to-one change of variable y = Pz shows that z*Mz is real and positive for any complex vector z if and only if y*Dy is real and positive for any y; in other words, if D is positive definite. For a diagonal matrix, this is true only if each element of the main diagonal—that is, every eigenvalue of M—is positive. Since the spectral theorem guarantees all eigenvalues of a Hermitian matrix to be real, the positivity of eigenvalues can be checked using Descartes' rule of alternating signs when the characteristic polynomial of a real, symmetric matrix M is available.

Klaas van Aarsen

MHB Seeker
Staff member
Sorry, but It seems straight to me.
Okay... so we're done?
I'm a bit confused since it's not clear to me what you are or were looking for.

MrJava

New member
Okay... so we're done?
I'm a bit confused since it's not clear to me what you are or were looking for.
Thank you, I think now I need to find my way to understand the explanation from Wiki then